Coalgebra

In mathematics, coalgebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras.

In finite dimensions, the duality is closer (see below).

Coalgebras occur naturally in a number of contexts (for example, group schemes).

There are also F-coalgebras, a category-theoretic generalisation of coalgebras, with important applications in computer science.

Formal definition

Formally, a coalgebra over a field K is a K-vector space C together with K-linear maps $Delta : C to C otimes_K C$ and $epsilon : C to K$ such that

$(mathrm{id}_C otimes Delta) circ Delta = (Delta otimes mathrm{id}_C) circ Delta$
$(mathrm{id}_C otimes epsilon) circ Delta = mathrm{id}_C = (epsilon otimes mathrm{id}_C) circ Delta$ .

(Here $otimes$ and $otimes_K$ refer to the tensor product over K.)

Equivalently, the following two diagrams commute:

In the first diagram we silently identify $Cotimes (Cotimes C)$ with $(C otimes C)otimes C$ ; the two are naturally isomorphic. Similarly, in the second diagram the naturally isomorphic spaces $C$ , $Cotimes K$ and $Kotimes C$ are identified.

The first diagram is the dual of the one expressing associativity of algebra multiplication (called the coassociativity of the comultiplication); the second diagram is the dual of the one expressing the existence of a multiplicative identity. Accordingly, the map Δ is called the comultiplication (or coproduct) of C and ε is the counit of C.

Examples

Take an arbitrary set S and form the K-vector space with basis S. The elements of this vector space are those functions from S to K that map all but finitely many elements of S to zero; we identify the element s of S with the function that maps s to 1 and all other elements of S to 0. We will denote this space by C. We define

Delta(s) = sotimes s quad mbox{ and } quad epsilon(s)=1 quad mbox{ for all } sin S.

By linearity, both Δ and ε can then uniquely be extended to all of C. The vector space C becomes a coalgebra with comultiplication Δ and counit ε (checking this is a good way to get used to the axioms).

As a second example, consider the polynomial ring K[X] in one indeterminate X. This becomes a coalgebra if we define

Delta(X^n) = sum_{m=0}^n X^motimes X^{n-m}

and

epsilon(X^n)=begin{cases}1& mbox{if } n=0

0& mbox{if } n>0 end{cases}

for all $nge 0.$ Again, because of linearity, this suffices to define Δ and ε uniquely on all of K[X]. Now K[X] is both a unital associative algebra and a coalgebra, and the two structures are compatible. Objects like this are called bialgebras, and in fact most of the important coalgebras considered in practice are bialgebras. Examples include Hopf algebras and Lie bialgebras.

In some cases the singular homology of a topological space form a coalgebra. lecture notes for reference

Finite dimensions

In finite dimensions, the duality between algebras and coalgebras is closer: the dual of a finite-dimensional (unital associative) algebra is a coalgebra, while the dual of a finite-dimensional coalgebra is a (unital associative) algebra. Indeed in general, the dual of a coalgebra is an algebra, but the dual of an algebra is not always a coalgebra.

The key point is that in finite dimensions, $(Aotimes A)^* cong A^*otimes A^*$ .

To distinguish these: in general, algebra and coalgebra are dual notions (meaning that their axioms are dual: reverse the arrows), while for finite dimensions, they are dual objects (meaning that a coalgebra is the dual object of an algebra and conversely).

If A is a finite-dimensional unital associative K-algebra, then its K-dual A^* consisting of all K-linear maps from A to K is a coalgebra. The multiplication of A can be viewed as a linear map $Aotimes Arightarrow A$ , which when dualized yields a linear map $A^*rightarrow (Aotimes A)^*$ . In the finite-dimensional case, $(Aotimes A)^*$ is naturally isomorphic to $A^*otimes A^*$ , so we have defined a comultiplication on A^*. The counit of A^* is given by evaluating linear functionals at 1.

Sweedler notation

When working with coalgebras, a certain notation for the comultiplication simplifies the formulas considerably and has become quite popular. Given an element c of the coalgebra (C,Δ,ε), we know that there exist elements c₍₁₎⁽ⁱ⁾ and c₍₂₎⁽ⁱ⁾ in C such that

Delta(c)=sum_i c_{(1)}^{(i)}otimes c_{(2)}^{(i)}.

In Sweedler's notation, this is abbreviated to

Delta(c)=sum_{(c)} c_{(1)}otimes c_{(2)}.

The fact that ε is a counit can then be expressed with the following formula

c=sum_{(c)} varepsilon(c_{(1)})c_{(2)} = sum_{(c)} c_{(1)}varepsilon(c_{(2)}).;

The coassociativity of Δ can be expressed as

sum_{(c)}c_{(1)}otimesleft(sum_{(c_{(2)})}(c_{(2)})_{(1)}otimes (c_{(2)})_{(2)}right) = sum_{(c)}left(sum_{(c_{(1)})}(c_{(1)})_{(1)}otimes (c_{(1)})_{(2)}right) otimes c_{(2)}.

In Sweedler's notation, both of these expressions are written as

sum_{(c)} c_{(1)}otimes c_{(2)}otimes c_{(3)}.

Some authors omit the summation symbols as well; in this sumless Sweedler notation, we may write

Delta(c)=c_{(1)}otimes c_{(2)}

and

c=varepsilon(c_{(1)})c_{(2)} = c_{(1)}varepsilon(c_{(2)}).;

Whenever a variable with lowered and parenthesized index is encountered in an expression of this kind, a summation symbol for that variable is implied.

Further concepts and facts

A coalgebra $(C,Delta,epsilon)$ is called co-commutative if $sigmacircDelta = Delta$ ;, where $sigma: Cotimes C to Cotimes C$ is the K-linear map defined by $sigma(cotimes d) = dotimes c$ for all c,d in C. In Sweedler's sumless notation, C is co-commutative if and only if

c_{(1)}otimes c_{(2)}=c_{(2)}otimes c_{(1)};

for all c in C. (It's important to understand that the implied summation is significant here: we are not requiring that all the summands are pairwise equal, only that the sums are equal, a much weaker requirement.)

If $(C_1,Delta_1,epsilon_1)$ and $(C_2,Delta_2,epsilon_2)$ are two coalgebras over the same field K, then a coalgebra morphism from $C_1$ to $C_2$ is a K-linear map $f:C_1to C_2$ such that $(fotimes f)circDelta_1 = Delta_2circ f$ and $epsilon_2circ f = epsilon_1$ . In Sweedler's sumless notation, the first of these properties may be written as:

f(c_{(1)})otimes f(c_{(2)})=f(c)_{(1)}otimes f(c)_{(2)}.

The composition of two coalgebra morphisms is again a coalgebra morphism, and the coalgebras over K together with this notion of morphism form a category.

A linear subspace I in C is called a coideal if I⊆ker(ε) and Δ(I)⊆I⊗C + C⊗I. In that case, the quotient space C/I becomes a coalgebra in a natural fashion.

A subspace D of C is called a subcoalgebra if Δ(D)⊆D⊗D; in that case, D is itself a coalgebra, with the restriction of ε to D as counit.

The kernel of every coalgebra morphism f : C₁ → C₂ is a coideal in C₁, and the image is a subcoalgebra of C₂. The common isomorphism theorems are valid for coalgebras, so for instance C₁/ker(f) is isomorphic to im(f).

As we have seen above, if A is a finite-dimensional unital associative K-algebra, then A^* is a finite-dimensional coalgebra, and indeed every finite-dimensional coalgebra arises in this fashion from some finite-dimensional algebra (namely from the coalgebra's K-dual). Under this correspondence, the commutative finite-dimensional algebras correspond to the cocommutative finite-dimensional coalgebras. So in the finite-dimensional case, the theories of algebras and of coalgebras are dual; studying one is equivalent to studying the other. However, things diverge in the infinite-dimensional case: while the K-dual of every coalgebra is an algebra, the K-dual of an infinite-dimensional algebra need not be a coalgebra.

Every coalgebra is the sum of its finite-dimensional coalgebras, something that's not true for algebras. In a certain sense then, coalgebras are generalizations of (duals of) finite-dimensional unital associative algebras.

Corresponding to the concept of representation for algebras is a corepresentation of a coalgebra. The notion is dual to the usual algebra representation as follow: An n-dimensional (left) corepresentation of C is given by a map $Pi:Vto Cotimes V$ where V is an n-dimensional vector space, such that it satisfies:

(Deltaotimes id)circPi=(idotimesPi)circPi

(epsilonotimes id)circPi=id

(The right corepresentation is defined similarly)

It can be represented as $Pi(e_i)=sum u_{ij}otimes e_j$ where $e_i$ is a basis of V, and $u=(u_{ij})$ is a matrix with elements in C (i.e. $u in M_n(C)$ ). The conditions above implies that